Energy release rate

Knowledge of the frequencies of highly explosive, moderately explosive, and nonexplosive eruptions would be useful in a variety of volcano studies. Historical records are generally incomplete, however, and contain very little quantitative data from which explosive magnitude can be estimated. Only the largest eruptions have a complete record back to the early 19th Century; other important explosive events went unrecorded prior to about 1960. Only a handful of the very biggest eruptions are represented in the geologic record, so it will be impossible to augment historical records post facto. A composite estimate of the magnitude of past explosive eruptions, termed the Volcanic Explosivity Index (VEI), is proposed as a semiquantitative compromise between poor data and the need in various disciplines to evaluate the record of past volcanism. The VEI has been assigned to over 8000 historic and prehistoric eruptions, and a complete list is available in a companion document. Tsuya, H., Geological and petrological studies of volcano Fuji,

In this chapter we consider reduced basis (RB) approximation and a posteriori error estimation for linear functional outputs of affinely parametrized linear parabolic partial differential equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametrically induced manifolddimension reduction; efficient and effective POD-Greedy RB sampling methods for identification of optimal and numerically stable approximations spaces -rapid convergence; rigorous and sharp a posteriori error bounds for the linear-functional outputs of interest -certainty; and Offline-Online computational decomposition strategies -minimum marginal cost. The RB approach is effective in the real-time context in which the expensive Offline stage is deemed unimportant -for example parameter estimation and control; the RB approach is also effective in the many-query context in which the expensive Offline stage is asymptotically negligible -for example design

The fracture toughnesses associated with fibre tensile failure and compressive fibre kinking in a T300/913 carbon-epoxy laminated composite are measured using compact tension and ‘compact compression’ tests respectively. The specimen strain fields were monitored using a digital speckle photogrammetry system during the tests. The damage present in the specimens after the tests was investigated using C-scan and optical and scanning electron microscopy. The initiation and propagation values of the tensile fibre failure critical energy release rate were determined as 91.6 kJ/m2 and 133 kJ/m2 respectively. For fibre compressive kinking, an initiation value of 79.9 kJ/m2 was obtained, but no meaningful propagation values could be determined. In both cases, the test results showed low scatter.

The finite element simulation of a selection of two-and three-dimensional flow problems is presented, based upon the use of four different constitutive models for polymer melts (Oldroyd-B, Rolie-Poly, Pom-Pom and XPP). The mathematical and computational models are first introduced, before their application to a range of visco-elastic flows is described. Results demonstrate that the finite element models used here are able to re-produce predictions made by other published numerical simulations and, significantly, by carefully conducted physical experiments using a commercial-grade polystyrene melt in a three-dimensional contraction geometry. The paper also presents a systematic comparison and evaluation of the differences between two-and threedimensional simulations of two different flow regimes: flow of an Oldroyd-B fluid around a cylinder and flow of a Rolie-Poly fluid into the contraction geometry. This comparison allows new observations to be made concerning the relatively poor quality of two-dimensional simulations for flows in even quite deep channels. This paper describes the finite element simulation of a variety of two-and 2 three-dimensional flows of polymer melts, based upon the use of a selection of 3 single-and multi-mode constitutive models. In particular we investigate the 4 differences between two-and three-dimensional simulations as the channel 5 depth varies and compare predictions against carefully conducted experi-6 ments. The primary goals of the work are to increase our understanding 7 of three-dimensional effects of visco-elastic fluids in deep channels and to 8 demonstrate that the constitutive models that we use are capable of repro-9 ducing physical experiments conducted with a commercial-grade polystyrene 10 flowing into a contraction. The computational approach used is based upon 11 a finite element discretization that uses equal order approximations for each 12 of the dependent variables. The stability of this discretization is achieved 13 through a slight relaxation of the incompressibility condition based upon a 14 penalty formulation. The advantages and disadvantages of such an approach 15 are discussed, and its accuracy is verified via comparison with other published 16 results. 17 In recent years there have been important developments in the numerical 18 2 simulation of the flow of molten polymers. New constitutive equations that 19 describe the nonlinear dynamics of entangled polymer have been developed, 20 for example [36, 38, 37, 32], based upon the Doi-Edwards tube model for 21 entanglements. These models incorporate approximations of the molecular 22 scale processes of reptation and chain-stretch, with parameters that relate to 23 the molecular structure of polymers although, in practice, these parameters 24 are often fitted to rheological measurements. 25 In particular multimode versions of the Pompom model [37] are able to 26 reproduce the shear and extensional rheology of branched polymer melts. 27 The polymeric stress in this model is factorised into contributions from tube 28 orientation and backbone stretch that are governed by separate equations. 29 Although Wapperom and Keunings [54] computed the full integral model for 30 the tube orientation, most simulations have used a differential approximation 31 that uses an Oldroyd B equation for the orientation distribution. 32 Unlike for the Oldroyd B model, the polymer stress in the Pompom model 33 is bounded, so that this model does not have the same high Weissenberg 34 number problem [7]. However, the polymer stretch equation contains a con-35 ditional statement that the stretch cannot exceed the arm number q, which is 36 difficult to handle numerically. To avoid this difficulty a number of modified 37 versions have been introduced that do not include this condition, these in-38 clude the XPP [52] and DCPP[11] models. The absence of the maximum con-39 dition significantly alters the behaviour in extension and removes the process 40 of arm retraction in the original model. However, recent extensional mea-41 surements on a melt of Pompom molecules [43] find greater strain-hardening 42 than is permitted by the maximum stretch condition in the Pompom model.

A model for the Homogeneous Charge Compression Ignition (HCCI) of Primary Reference Fuels (PRFs) in a Rapid Compression Machine (RCM) has been developed. A reduced chemical kinetic model that included 32 species and 55 reactions was used and the affect of wall heat transfer on the temperature of the adiabatic core gas was taken into account by adding the displacement volume of the laminar boundary layer to the cylinder volume. A simple interaction between n-heptane and iso-octane was also included. The results showed the well-known two-stage ignition characteristics of heavy hydrocarbons, which involve low and high temperature cycles followed by a branched chain explosion. The first stage energy release decreases and the ignition delay increases nonlinearly with increasing octane number and decreasing the initial pressure. The energy release rate and total energy released were determined primarily by the rate of CO oxidation during the explosive phase following the ignition delay. The model reproduced the pressure curves obtained in the RCM experiments over a wide range of conditions remarkably well and was very sensitive to the fuel structure, the mixture composition and the initial temperature and pressure. Thus, the model can be easily adapted for predicting "knock" in spark-ignition engines and ignition-delays and burning rates in HCCI engines.

A generalized potential-based constitutive model for mixed-mode cohesive fracture is presented in conjunction with physical parameters such as fracture energy, cohesive strength and shape of cohesive interactions. It characterizes different fracture energies in each fracture mode, and can be applied to various material failure behavior (e.g. quasi-brittle). The unified potential leads to both intrinsic (with initial slope indicators to control elastic behavior) and extrinsic cohesive zone models. Path dependence of work-of-separation is investigated with respect to proportional and non-proportional paths-this investigation demonstrates consistency of the cohesive constitutive model. The potential-based model is verified by simulating a mixed-mode bending test. The actual potential is named PPR (Park-Paulino-Roesler), after the first initials of the authors' last names.

We have used observed band-pass filtered accelerograms and a previously determined slip distribution to invert for the dynamic rupture propagation of the 1992 Landers earthquake. In our simulations, dynamic rupture grows under the simultaneous control of initial stress and rupture resistance by friction, which we modeled using a simple slip-weakening law. We used a simplified Landers fault model where the fault segments were combined into a single vertical, planar fault. By trial and error we modified an initial stress field, inferred from the kinematic slip distribution proposed by Wald and Heaton [1994], until dynamic rupture generated a rupture history and final slip distribution that approximately matched those determined by the kinematic inversion. We found that rupture propagation was extremely sensitive to small changes in the distribution of prestress and that a delicate balance with energy release rate controls the average rupture speed. For the inversion we generated synthetic 0.5 Hz ground displacements using an efficient Green's function propagator method (AXITRA). This method enables us to propagate the radiation generated by the dynamic rupture to distances greater than those feasible using the finite difference method. The dynamic model built by trial-and-error inversion provides a very satisfactory fit between synthetics and strong motion data. We validated this model using records from stations used in the slip inversion as well as some which were not included. We also inverted for a complementary model that fits the data just as well but in which the initial stress was perfectly uniform while rupture resistance was heterogeneous. This demonstrates that inversion of ground motion is nonunique.

The way that faults transport crustal fluids is important in many fields of earth sciences such 15 as petroleum geology, geothermal research, volcanology, seismology, and hydrogeology. For 16 understanding the permeability evolution and maintenance in a fault zone, its internal 17 structure and associated local stresses and mechanical properties much be known. This 18 follows because the permeability is primarily related to fracture propagation and their linking 19 up into interconnected clusters in the fault zone. Here we show that a fault zone can be 20 regarded as an elastic inclusion with mechanical properties that differ from those of the host 21 rock. As a consequence, the fault zone develops its own local stresses which differ from the 22 associated regional stresses. The local stresses, together with fault-rock heterogeneities and 23 interfaces (discontinuities; fractures, contacts), determine fracture propagation, deflection 24 (along discontinuities/interfaces), and arrest in the fault zone and, thereby, its permeability 25 development. We provide new data on the internal structure of fault zones, in particular the 26 fracture frequency in the damage zone as a function of distance from the fault core. New 27 numerical models show that the local stress field inside a fault zone, modelled as an 28 inclusion, differ significantly from those of the host rock, both as regards the magnitude and 29 the directions of the principal stresses. Also, when the mechanical layering of the damage 30 zone, due to its variation in fracture frequency, is considered, the numerical models show 31 Manuscript Click here to view linked References 2 abrupt changes in local stresses not only between the core and the damage zone but also 32 within the damage zone itself. Abrupt changes in local stresses within the fault zone generate 33 barriers to fracture propagation and contribute to fracture arrest. Also, analytical solutions on 34 the effects of material toughnesses (critical energy release rates) of layers and their interfaces 35

It is shown that unless the substrate is at least as stiff as the film, the energy stored in the substrate contributes significantly to the energy release rate of film delamination under compression either with or without cracking. For very compliant substrates, such as polyethylene terephthalate (PET) with a indium tin oxide (ITO) film, the energy release rate allowing for the deformation of the substrate can be more than an order of magnitude greater than the value obtained neglecting the substrate's deformation. The argument that buckling delaminations tunnel at the tip rather than spread sideways because of increase in mode-mixity may need modification; it is still true for stiff substrates, but for compliant substrates the average energy release rate decreases with delamination width and the limitation in buckled width may be due to this stability as much as the increase in mode-mixity.

During needle-based procedures, transitions between tissue layers often lead to rupture events that involve large forces and tissue deformations and produce uncontrollable crack extensions. In this paper, the mechanics of these rupture events is described, and the effect of insertion velocity on needle force, tissue deformation, and needle work is analyzed. Using the J integral method from fracture mechanics, rupture events are modeled as sudden crack extensions that occur when the release rate J of strain energy concentrated at the tip of the crack exceeds the fracture toughness of the material. It is shown that increasing the velocity of needle insertion will reduce the force of the rupture event when it increases the energy release rate. A nonlinear viscoelastic Kelvin model is then used to predict the relationship between the deformation of tissue and the rupture force at different velocities. The model predicts that rupture deformation and work asymptotically approach minimum values as needle velocity increases. Consequently, most of the benefit of using a higher needle velocity can be achieved using a finite velocity that is inversely proportional to the relaxation time of the tissue. Experiments confirm the analytical predictions with multilayered porcine cardiac tissue.

An analysis is performed for the transient response of a semi-infinite, anti-plane crack propagating in a hexagonal piezoelectric medium. The mixed boundary value problem is solved by transform methods together with the Wiener-Hopf and Cagniardde Hoop techniques. As a special case, a closed form solution is obtained for constant speed crack propagation under external anti-plane shear loading with the conducting electrode type of electric boundary condition imposed on the crack surface (a second type of boundary condition is considered in Part II of this work). In purely elastic, transversely isotropic elastic solids, there is no antiplane mode surface wave. However, for certain orientations of piezoelectric materials, a surface wave will occur-the Bleustein-Gulyaev wave. Since surface wave speeds strongly influence crack propagation, the nature of antiplane dynamic fracture in piezoelectric materials is fundamentally different from that in purely elastic solids, exhibiting many features only associated with the in-plane modes in the elastic case. For a general distribution of crack face tractions, the dynamic stress intensity factor and the dynamic electric displacement intensity factor are derived and discussed in detail for the electrode case. As for inplane elastodynamic fracture, the stress intensity factor and energy release rate go to zero as the crack propagation velocity approaches the surface wave speed. However, the electric displacement intensity does not vanish.

A well-known four-point bending test has been modified such that the critical energy release rate G c for delaminating cracks propagating at the interface of a thin, brittle layer bonded to a substrate can be measured. The energy release rate G required for crack delaminating at those interfaces is obtained by attaching a stiffening layer to the layer system. Another advantage of this modification is that segmentation of the layer and plastic deformation of the substrate during bending are avoided. The interface fracture energy G c of a plasma sprayed ZrO 2-ceramic layer on a flame sprayed high alloyed steel substrate has been measured.

AbstractÐInterfacial adhesion is becoming a critical material property for improving the reliability of multilayer thin ®lm structures used in microelectronics. Cross-sectional nanoindentation (CSN) is a new mechanical test especially designed for measuring the fracture toughness of thin ®lm interfaces. Interfacial fracture is achieved by nanoindentation in the structure cross-section. A model based on the elastic plate theory has been developed to calculate numerically the interfacial critical energy release rate (G ci ) for cer-amic±ceramic systems from CSN test results. The model inputs are the thin ®lm elastic properties, thin ®lm thickness, interfacial crack area and maximum thin ®lm de¯ection during the test. Closed form analytical solutions, obtained for two limiting cases, are consistent with the numerical approach. This technique has been successfully applied to silicon nitride±silicon oxide thin ®lms, commonly used as electrical isolators in microelectronic devices. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

An improved method based on the ÿrst-order shear deformable plate theory is developed to calculate the energy release rate and stress intensity factor for a crack at the interface of a bi-layer structure. By modeling the uncracked region of the structure as two separate Reissner-Mindlin plates bonded perfectly along the interface, this method is able not only to take into account the shear deformation in the cracked region, but also to capture the shear deformation in the uncracked region of the structure. A closed form solution of energy release rate and mode decomposition at the interface crack is obtained for a general loading condition, and it indicates that the energy release rate and stress intensity factor are determined by two independent loading parameters. Compared to the approach based on the classical plate theory, the proposed method provides a more accurate prediction of energy release rate as well as mode decomposition. The computational procedures introduced are relatively straightforward, and the closed form solution can be used to predict crack growth along the layered structures. ?

Recent analyses of long duration event (LDE) Ñares indicate successive occurrences of magnetic reconnection and resultant energy release in the decay phase. However, quantitative studies of the energy release rate and the reconnection rate have not yet been made. In this paper we focus on the decay phase of an LDE Ñare on 1997 May 12 and derive the energy release rate H and the reconnection rate where is the inÑow velocity and is the velocity. For this purpose, we utilize a M A \ v in /v A , v in v A Alfve n method to determine and the coronal magnetic Ðeld indirectly, using the following relations : v in B corona

We study the propagation of seismic ruptures along a fault surface using a fourth-order finite difference program. When prestress is uniform, rupture propagation is simple but presents essential differences with the circular self-similar shear crack models of Kostrov. The best known is that rupture can only start from a finite initial patch (or asperity). The other is that the rupture front becomes elongated in the in-plane direction. Finally, if the initial stress is sufficiently high, the rupture front in the in-plane direction becomes super-shear and the rupture front develops a couple of ''ears'' in the in-plane direction. We show that we can understand these features in terms of single nondimensional parameter s that is roughly the ratio of available strain energy to energy release rate. For low values of s rupture does not occur because Griffith's criterion is not satisfied. A bifurcation occurs when s is larger than a certain critical value, s c. For even larger values of s rupture jumps to super-shear speeds. We then carefully study spontaneous rupture propagation along a long strike-slip fault and along a rectangular asperity. As for the simple uniform fault, we observe three regimes: no rupture for subcritical values of s, sub-shear speeds for a narrow range of supercritical values of s, and super-shear speeds for s\ 1.3s c. Thus, there seems to be a certain universality in the behavior of seismic ruptures.

To investigate possible indicators of critical point behavior prior to rock failure, the statistical properties of pre-failure damage were analyzed based on acoustic emission events (AE) recorded during the catastrophic fracture of typical rock samples under differential compression. AEs were monitored using a high-speed 32-channel waveform recording system. Time-dependent statistics, including the energy release rate, b-value of the magnitude-frequency distribution, fractal dimension and spatial correlation length (SCL) of the AE hypocenters were calculated for each data set. Each parameter is a function of the time-to-failure and thus can be treated as an indicator of the critical point. It is clear that the pre-failure damage evolution prior to catastrophic failures in several common rocktypes is generally characterized by: 1) accelerated energy release, 2) a decrease in fractal dimension and SCL with a subsequent precursory increase, and 3) a decrease in b-value from ∼ 1.5 to ∼ 0.5 for hard rocks, and from ∼ 1.1 to ∼ 0.8 for soft rocks such S-C cataclasite. However, each parameter also reveals more complicated temporal evolution due to either the heterogeneity of the rock mass or the micro-mechanics of shear fracturing. This confirms the potential importance of integrated analysis of two or more parameters for successfully predicting the critical point. The decreasing b-value and increasing energy release may prove meaningful for intermediate-term prediction, while the precursory increase in fractal dimension and SCL may facilitate short-term prediction.

The mode I and mode II fracture toughness and the critical strain energy release rate for different concrete-concrete jointed interfaces are experimentally determined using the Digital Image Correlation technique. Concrete beams having different compressive strength materials on either side of a centrally placed vertical interface are prepared and tested under three-point bending in a closed loop servo-controlled testing machine under crack mouth opening displacement control. Digital images are captured before loading (undeformed state) and at different instances of loading. These images are analyzed using correlation techniques to compute the surface displacements, strain components, crack opening and sliding displacements, load-point displacement, crack length and crack tip location. It is seen that the CMOD and vertical load-point displacement computed using DIC analysis matches well with those measured experimentally.

Three techniques of bond strength determination in micromechanical tests-fibre strain profile analysis by means of Raman spectroscopy, "kink" force determination in a traditional pull-out test, and crack length monitoring in a microbond test-were used for investigation of interfacial debonding in epoxy-glass fibre and epoxy-aramid fibre systems. Crack propagation was characterised by local interfacial parameters-critical energy release rate, G ic , and ultimate interfacial shear strength (IFSS), t ult . The comparison of the results showed good agreement both between different techniques and between stress-based and energy-based failure criteria. Sizing of glass fibres caused more pronounced variations in the IFSS than for aramid fibres due to different interfacial failure patterns. The strength of "real" epoxy-glass composites with sized and unsized fibres correlates well with the bond strength determined from the micromechanical tests. ᭧

The failure mechanisms operative in a thermal barrier system with a NiCoCrAlY bond coat are ascertained by examining test specimens subjected to thermal cycling in a burner rig. The findings are augmented by observations made on an actual turbine blade. The morphology and microstructure of the thermally grown oxide (TGO) have been characterized: emphasizing heterogeneities, especially ''pegs'', and the TGO thickness at failure, h tgo . In each case, a dominant delamination has been identified, that extends primarily along the interface between the TGO and the bond coat: a finding that contrasts with the multiple sub-critical cracks found when other bond coats are used. The principal stresses, r 1 , have been analyzed on a cycle-by-cycle basis, incorporating plastic deformation of the bond coat and TGO, as well as TGO growth. Energy release rates, G, for putative cracks at the interface have also been ascertained. These analyses reveal that r 1 and G are too small to nucleate interface cracks, except at occasional large edgeimperfections, such as vertical separations in the thermal barrier coating (TBC). Once formed, these cracks become unstable and delaminate the interface. Calculations of the delamination energy release rate, upon comparison with the interface toughness, a critical TGO thickness, ðh tgo Þ c % 3 lm, comparable to that found experimentally.

The simulation of the delamination process in composite structures is quite complex, and requires advanced FE modelling techniques. Failure analysis tools must be able to predict initiation, size and propagation of delamination process. The objective of the paper is to present modelling techniques able to predict a delamination in composite structures. Four different ways of modelling delamination growth of a double cantilever beam test (DCB) are proposed. The first two approaches were based on a cohesive zone model: the interface being represented either by using delamination elements or non-linear springs. The idea of the third approach was to use a fracture mechanics criterion, but to avoid the complex moving mesh techniques it often implies. The interface between the two layers was simulated with solid elements representing the matrix, which were eliminated when their energy release rate exceeded the critical value. The energy release rate was computed using the virtual crack closure technique (VCCT). In the last approach, the interface behaviour was modelled by a tiebreak contact. Coincident nodes were tied together with a constraint relation and remained joined, until when the maximum interlaminar stresses was reached. Once this value was exceeded, the nodes associated with that constraint were released to simulate the initiation of delamination. The comparison of the results of the first three modelling techniques with experimental data showed that very good correlation was achieved. Poor results were obtained using tiebreak contact. It was due to the criterion used, since when the critical interlaminar stress was reached, the delamination was experienced before the critical energy release rate was reached.

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).

d e n t a l m a t e r i a l s 2 6 ( 2 0 1 0 ) e63-e77 a v a i l a b l e a t w w w . s c i e n c e d i r e c t . c o m j o u r n a l h o m e p a g e : w w w . i n t l . e l s e v i e r h e a l t h . c o m / j o u r n a l s / d e m a Energy release rate a b s t r a c t Dental adhesives are usually tested in shear or tension even though neither of these approaches measures the local stress triggering failure. Because the stress level varies extensively over the bonded surface, it seems as a fracture mechanics approach would be more appropriate.

The so-called ''damage mesomodel for laminates'' developed in the past fifteen years, particularly at Cachan, is being reconsidered in the light of the recent works, theoretical as well as experimental, done on the microscale. Previous works have already proved that this mesomodel can be interpreted as the homogenized result of micromodels involving common microdamage mechanisms: transverse microcracking, delamination at the tips of transverse microcracks and fiber-matrix debonding. Here, we are going one step further by considering microcracking in skin plies of various thicknesses. To fit experimental results better, we introduced a modification of the coupling of the three microdamage scenarios in all plies. Contrary to the classical approach, the critical values of the energy release rate are not associated with a healthy material but with a damaged material; the first damage mechanism to occur is associated with nontransverse microcracks due to nearly uniformly distributed fiber-matrix debonding in the damaged ply. These mechanisms and this coupling are homogenized on the mesoscale, which leads to an improved version of the ''damage mesomodel for laminates'' involving only a small number of material constants which can be interpreted on the microscale. Thickness effects are taken into account: the improved version, unlike the previous one, is valid for arbitrary values of the thickness. We discuss experimental data and identification using carbon-epoxy laminates at room temperature.

Several criteria for interface fracture are examined and compared to test results obtained from glass/epoxy specimens. These include two energy release rate criteria, a critical hoop stress criterion and a critical shear stress criterion. In addition, approximate plastic zone size and shape within the epoxy are determined for these tests.

Compaction tests (isotropic drained loading) were conducted on randomly packed glass beads that were: 1) uncemented and 2) cemented by epoxy at their contacts. In the latter case, the volume of the epoxy accounted for 10% of the pore space. Intensive crushing of grains was observed in the first case at about 50 MPa. In the second case, the cemented grains stayed intact, the failure being localized within the epoxy. Therefore, even small amounts of cement at contacts prevent the failure of grains. Theoretically, this effect follows from the theory of cemented granular materials: stress concentration is high at the contacts of uncemented grains, whereas even small amounts of relatively soft cement result in a more uniform stress distribution over a larger contact area. (Authors) 951052

Noether's theorem on invariant variational principles is applied in the case of in®nitesimal couple stress elasticity, thereby extending the analysis of Knowles and Sternberg (1972. On a class of conservation laws in linearized and ®nite elastostatics. Arch. Ration. Mech. Anal. 44, 187±211) beyond the range of classical elasticity. Two conserved integral quantities are deduced which generalize the J-integral and L-integral in the notation of Budiansky and Rice (1973: Budiansky, B. and Rice, J. R. (1973) Conservation laws and energy-release rates. J. Appl. Mech. 40, 201±203). An expression for an M-integral is also obtained, but it is demonstrated that there is no corresponding conservation law for this integral. Relationships of the derived path integrals to other similar quantities for couple stress elasticity which have appeared in the literature are discussed.

The characterization of critical energy release rates of adhesive joints in laminated composite structures is a key issue when failure analyses have to be performed. Critical energy release rates, or fracture toughnesses, are known to be dependent on the mode mixing ratio, i.e. the portion of shear loading. It is thus useful to determine a criterion which gives the critical energy release rate as a function of the mode mixing ratio, which is the overall goal of this paper. For this purpose several experiments have been performed, for single mode I, single mode II, and mixed mode I/II loading conditions with pre-defined mode mixing ratios. Unfortunately, most of the experimental outcome cannot be used directly for least squares fitting of suitable fracture toughness criteria due to a couple of reasons, which will be discussed in detail. Hence, a numerical approach based on cohesive interface elements is employed to determine some of the critical energy release rates by fitting against experimental load-deformation curves. This combined numerical/experimental approach yields a useful database of discrete critical energy release rate values. These are utilized to fit suitable criteria which then allow the calculation of critical energy release rates for any given mode mixing ratio. The results are discussed in terms of convergence to the discrete values and physical plausibility, and a simple possibility to include mode III behavior is presented. 1

The interlaminar fracture behavior of a unidirectionally glass ®ber reinforced composite under the full range of in-plane loading conditions has been investigated. Loading conditions from pure mode I through various mixed mode I/II ratios up to pure mode II have been generated by the aid of the proposed compound version of the CTS (compact tension shear) specimen. From the experimentally measured critical loads, the mode I, mode II and the various mixed mode I/II critical energy release rates at crack initiation have been determined by the aid of the ®nite element method and the modi®ed virtual crack closure integral method. Based on these results the parameters for a fracture criterion for the composite under consideration have been determined. # Engineering Fracture Mechanics 61 (1998) 325±342 0013-7944/98/$ -see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 -7 9 4 4 ( 9 8 ) 0 0 0 6 8 -X PERGAMON fracture behaviour of composite materials. The double cantilever beam (DCB) and the end notched¯exure (ENF) specimens have been employed for mode I and mode II tests, respectively. Also for the mixed mode tests, mostly beam type specimens were used in order to obtain mixed mode I/II critical energy release rates.

The path-independent J Ã k -integral, in conjunction with the finite element method (FEM), is presented for mode I and mixed-mode crack problems in orthotropic functionally graded materials (FGMs) considering plane elasticity. A general procedure is presented where the crack is arbitrarily oriented, i.e. it does not need to be aligned with the principal orthotropy directions. Smooth spatial variations of the independent engineering material properties are incorporated into the element stiffness matrix using a ''generalized isoparametric formulation'', which is natural to the FEM. Both exponential and linear variations of the material properties are considered. Stress intensity factors and energy release rates for pure mode I and mixed-mode boundary value problems are numerically evaluated by means of the equivalent domain integral especially tailored for orthotropic FGMs. Numerical results are discussed and validated against available theoretical and numerical solutions.

Coatings subject to residual compression eventually fail by buckle-driven delamination. The phenomenon is most vivid in thermal barrier coatings (TBCs) used in gas turbines. The failure evolution commences with the formation of a large number of small cracks at geometric imperfections near the interface. These cracks spread upon thermal exposure, particularly upon thermal cycling, because of the formation of a thermally grown oxide (TGO) beneath the TBC, which introduces normal and shear stress near the interface. Experimental observations indicate that some of these cracks coalesce to form large-scale delaminations susceptible to buckling. The mechanics governing crack coalescence and the consequent failure are addressed in the present analysis. A model is introduced that simulates stresses induced in the TBC by spatial variations in TGO growth. Energy release rates for cracks evolving in this stress field are determined. Two related scenarios are considered, which differ in the way the TGO shape evolves. In both, contact between the crack faces and the consequent wedging action is responsible for ultimate coalescence. The wedging force induces a mode I stress intensity that becomes infinite as the cracks coalesce. The consequence is that, for some TGO shapes, the energy release rate is always non-zero, with a minimum at a characteristic crack length. This minimum establishes a criterion for crack coalescence and failure. Based on these insights, finite element simulations have been used to predict cyclic crack growth rates in a TBC system that correlate well with experimental observations.

Biomass characterization and fire behavior documentation were carried out on two large (>2000 ha) experimental fires conducted in arid savanna fuels in Kruger National Park in September 1992. Prefire fuel loads, fuel consumption, spread rates, flame zone characteristics, and in-fire and perimeter wind field dynamics were measured in order to determine overall energy release rates for each fire. Convection column dynamics were also measured in support of airborne trace gas and particulate measurements. Energy release rates varied significantly between the two fires, and this was strongly reflected in convection column development. The lower-intensity fire produced a weak, poorly defined smoke plume, while a well-developed column with a capping cumulus top developed during the higher intensity fire. Further experimental burning studies, in savannas with higher fuel loads, are recommended to further explore the fire behavior-convection column dynamics relationship investigated in this study.

By using the method presented by Isobe et al. (2002), the non-dimensional reconnection rate v in /v A has been determined for the impulsive phase of three two-ribbon flares, where v in is the velocity of the reconnection inflow and v A is the Alfvén velocity. The non-dimensional reconnection rate is important to make a constraint on the theoretical models of magnetic reconnection. In order to reduce the uncertainty of the reconnection rate, it is important to determine the energy release rate of the flares from observational data as accurately as possible. To this end, we have carried out one dimensional hydrodynamic simulations of a flare loop and synthesized the count rate detected by the soft X-ray telescope (SXT) aboard Yohkoh satellite. We found that the time derivative of the thermal energy contents in a flare arcade derived from SXT data is smaller than the real energy release rate by a factor of 0.3-0.8, depending on the loop length and the energy release rate. The result of simulation is presented in the paper and used to calculate the reconnection rate. We found that reconnection rate is 0.047 for the X2.3 flare on 2000 November 24, 0.015 for the M3.7 flare on 2000 July 14, and 0.071 for the C8.9 flare on 2000 November 16. These values are similar to that derived from the direct observation of the reconnection inflow by Yokoyama et al. (2001), and consistent with the fast reconnection models such as that of Petschek (1964).

This paper focuses on aspects of fracture mechanics of interface cracks in anisotropic bimaterials. In this case there is a coupling of all three crack-tip fracture modes in the natural interface-crack coordinate system, whereas in the isotropic case, mode 3 is decoupled from modes 1 and 2. This paper intends to shed light on how to interpret the crack-tip fields which are given explicitly along the interface in terms of two (real 3x3) bimaterial matrices W and D. A matric function Y is defined in terms of W and D which determines the coupling and oscillations in the cracktip fields. Explicit expressions for the crack-tip fields and the associated stress intensity factors are given as well as for the energy release rate. The finite (Griffith) interface crack is considered in detail.

A new finite element technique for calculating energy release rates is presented. An explicit expression for energy changes due to virtual crack extensions is formulated based on a variation of isoparametric element mappings. Energy release rates are calculated directly from integral expressions evaluated over singular quarter-point isoparametric elements surrounding the crack tip. Since the energy release rates are expressed in variational form, there is no need for the analyst to select a small finite crack extension to simulate a virtual crack extension. The method is shown to produce very accurate solutions even with fairly coarse element meshes. A similar technique for mixed-mode fracture based on mutual potential energy release rates is described.

The recently developed edge-based smoothed finite element method (ES-FEM) is extended to the mixmode interface cracks between two dissimilar isotropic materials. The present ES-FEM method uses triangular elements that can be generated automatically for problems even with complicated geometry, and strains are smoothed over the smoothing domains associated with the edges of elements. Considering the stress singularity in the vicinity of the bimaterial interface crack tip is of the inverse square root type together with oscillatory nature, a five-node singular crack tip element is devised within the framework of ES-FEM to construct singular shape functions. Such a singular element can be easily implemented since the derivatives of the singular shape term (1/ √ r ) are not needed. The mix-mode stress intensity factors can also be easily evaluated by an appropriate treatment during the domain form of the interaction integral. The effectiveness of the present singular ES-FEM is demonstrated via benchmark examples for a wide range of material combinations and boundary conditions.

A recently developed eigenfunction expansion technique, based in part on separation of the thickness-variable, is first developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness crack/anticrack weakening/ reinforcing an infinite monoclinic plate, of finite thickness and subjected to far-field antiplane shear loading. Crack/anticrack-face boundary conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the anisotropic (monoclinic) plate are exactly satisfied. Five different through-thickness crack/anticrack-face boundary conditions are considered: (i) slit crack, (ii) anticrack or perfectly bonded rigid inclusion, (iii) transversely rigid inclusion (longitudinal slip permitted), (iv) rigid inclusion in part perfectly bonded, the remainder with slip, and (v) rigid inclusion located alongside a crack. The three-dimensional stress intensity factor for a center-crack, and stress singularity coefficients for/on a center-anticrack are then derived by incorporating an extension of the Stroh type approach in the present analysis. Through-thickness distribution of stress intensity factor and stress singularity coefficient for a crack and an anticrack, respectively, is also presented. Additionally, singular residual stress fields in the vicinity of the fronts of these cracks, anticracks and similar discontinuities are also discussed. Hitherto unavailable expressions for three-dimensional energy release and absorption rates for center-cracks and anticracks are derived by using Irwin's crack closure and Eshelby's eigenstrain approach, respectively. A heretofore unavailable expression for the energy release rate on the super-rigid inclusion is derived, using an approach which is analogous, in a reverse sense, to Irwin's crack closure method. Finally, a new mode III crack deflection/bifurcation criterion is also derived. The crack deviation under antiplane shear loading is strongly correlated with the elastic stiffness constant, c 45 , of the monoclinic single crystal or off-axis composite lamina concerned.

The Boeing sol-gel conversion coating (Boegel-EPII), derived from an acidcatalyzed aqueous solution of organofunctional silane and zirconium alkoxide precursors, is being used as an adhesion promoter for adhesive bonding and painting applications in the aerospace industry. A unique advantage of the sol-gel process is that strong and durable bonds are produced without the hazardous chemical usage and rinse-water requirements of conventional anodizing or etching processes. In this study, a fracture mechanics method was used to investigate the adhesion properties of sol-gel-reinforced epoxy=aluminum joints. The Hugh Brown asymmetric double cantilever beam (ADCB) wedge test was employed, which allowed the measurements of the critical energy-release rate, subcritical crack-growth kinetics, and threshold energy-release rate on a single sample in a reasonably short period of time. These experiments were carried out with aluminum substrates on which the surface morphology was systematically varied by polishing, sanding, grit-blasting, and chemical etching. X-ray photoelectron spectroscopy (XPS) and scanning electron microscopy (SEM) were used to identify the locus of failure. The surface morphology of the substrates was characterized with SEM, optical profilometry, and spreading kinetics. The macrorough structures drive the crack to within a thin epoxy layer close to the polymer=metal interface, which enhances the initial strength of the sol-gel-reinforced interface. The microroughness of the substrate is, however, more effective than the 487 macroroughness in enhancing the durability. Lastly, an attempt has been made to correlate the energy-release rate with the fractal dimension for sol-gel-reinforced joints with macrorough substrates.

A simplified mixed-mode fracture analysis combining nonlinear thin-plate stress solutions with crack-tip elasticity results has been developed to account for local variations of G~, GH and G~H in thin-film debond problems associated with large film deformations. Membrane and bending stresses from the plate analysis are matched with the crack-tip singularity solution over a small boundary region at the crack tip where the effect of geometric nonlinearity is small. Local variations in each of the individual components of the energy release rate are directly related to the "jump" in these stresses across the crack border. Specific results are presented for 1-D and elliptical planeform cracks. Deformations were induced either by a pressure acting normal to the film surface or biaxial compression or tension stresses applied to the substrate in which the loading axes and debond axes coincide. The latter type of loading involves buckling of the delaminated film. The model predictions compare well with more rigorous solutions provided the film thickness is small compared to the debond dimensions. In all cases analyzed, G~I ~ was negligible. The ratio G~/GH typically decreases with increasing load or film deformation, the rate was moderate for pressure loading while generally sharp for compression loading. Film-substrate overlap may occur for certain debond geometry and loading conditions. Prevention of this by the substrate may critically increase the energy available for crack propagation.

A layerwise B-spline finite strip method is developed with consideration of delamination kinematics to study the buckling, postbuckling and delamination propagation in debonded composite laminates under compression in Part 1 [Zhang Y, Wang S. Buckling, post-buckling and delamination propagation in debonded composite laminates Part 1: Theoretical development. Compos Struct 2009;88:121-30]. Extensive numerical applications are conducted in this Part 2 to validate the theoretical development. Comparisons are made between the present predictions and the existing numerical and experimental results.

AbstnC't-A procedure of total energy release rate and stress intensity factors is developed for general non-homogeneous laminated composite laminates. The total energy release rate is obtained by using the J-integral for a one dimensional model of plane stress, plane strain and cylindrical bending. Decomposition of it into mode I and mode II, by which the mode mixity calculation is carried out, is based on the assumption of equivalent orthotropic properties through the laminate thickness. The process is straightforward and can be used as a criterion for delamination onset and growth of one dimensional structural model under general loading in the pre-and post-buckling states.

An accurate laminate model developed by using multilayer shear deformable plate modeling and interface elements, based on fracture mechanics and contact mechanics, is proposed to analyze mixed mode delamination in composite laminates. Perfect adhesion along the undelaminated portion of the delamination plane is simulated by treating interface stiffnesses as penalty parameters, whereas to enforce interface displacement continuity between plate elements constituting each sub-laminate above or below the delamination plane, the Lagrange multiplier method is used. The governing differential equations are derived through a variational procedure by using a modified total potential energy functional. Results are obtained by numerical integration of the non-linear three-point boundary value problem modeling mixed-mode delamination of the laminate plate subjected to end loading, which accounts also for the frictionless contact condition.

We present the relation of the spatial distribution of Hα kernels with the distribution of hard X-ray (HXR) sources seen during the 2001 April 10 solar flare. This flare was observed in Hα with the Sartorius telescope at Kwasan Observatory, Kyoto University, and in hard X-rays (HXRs) with the Hard X-ray Telescope (HXT) onboard Yohkoh. We compared the spatial distribution of the HXR sources with that of the Hα kernels. While many Hα kernels are found to brighten successively during the evolution of the flare ribbons, only a few radiation sources are seen in the HXR images. We measured the photospheric magnetic field strengths at each radiation source in the Hα images, and found that the Hα kernels accompanied by HXR radiation have magnetic strengths about 3 times larger than those without HXR radiation. We also estimated the energy release rates based on the magnetic reconnection model. The release rates at the Hα kernels with accompanying HXR sources are 16-27 times larger than those without HXR sources. These values are sufficiently larger than the dynamic range of HXT, which is about 10, so that the difference between the spatial distributions of the Hα kernels and the HXR sources can be explained.

An interface crack with a frictionless contact zone at the right crack-tip between two dissimilar magnetoelectroelastic materials under the action of concentrated magnetoelectromechanical loads on the crack faces is considered. The open part of the crack is assumed to be magnetically impermeable and electrically permeable. The Dirichlet-Riemann boundary value problem is formulated and solved analytically. Stress, magnetic induction and electrical displacement intensity factors as well as energy release rate are thus found in analytical forms. Analytical expressions for the contact zone length have been derived. Some numerical results are presented and compared with those based on the other crack surface conditions. It is shown clearly that the location and magnitude of the applied loads could significantly affect the contact zone length, the stress intensity factor and the energy release rate.

A first-order shear deformable plate theory-based method is developed to calculate the strain energy release rate and stress intensity factor of a non-homogeneous delaminated composite plate under a general three-dimensional (3-D) loading condition. By modeling the delaminated plate as two shear deformable sub-laminates on either side of the delaminated plane, the strain energy release rate is expressed in terms of three concentrated forces at the crack tip and their corresponding compliance coefficients. The simple expression of strain energy release rate makes the mode decomposition under complicated loading situation possible with the aid of two supplementary continuum analyses. To illustrate the present method, a plain strain delamination problem of laminates is examined, and the closed-form expressions of strain energy release rate and stress intensity factor are obtained. It is found that the available solutions, such as the ones based on the classical plate theory, can be recovered from the present solutions by simply neglecting the transverse shear force. The relationship between the global and local decompositions is further established, and the accuracy of the present solutions is examined by comparing with numerical results of boundary element method. With inclusion of transverse shear deformation in the formulation, more accurate and explicit predictions of the strain energy release rate and stress intensity factor of delaminated composite plate are achieved by the present method, especially when a laminate has a relatively low transverse shear modulus or moderate thickness.

The topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced at a point of the domain. In this fracture mechanics work we use the topological derivative to approximate the energy release rate field associated with a small crack at any boundary location and at any orientation. Our proposed method offers significant computational advantages over current finite element based methods since it requires a single analysis, whereas the others require a distinct analysis for each crack location-orientation combination. Moreover, the proposed method evaluates the topological derivative in the non-cracked domain which eliminates the need for tailored meshes in the crack region. & 2011 Elsevier Ltd. All rights reserved. associated with a small crack at any boundary location and at any orientation in a linear elastic isotropic two-dimensional (2-D) domain. The topological derivative field indicates the variation of a response functional when an infinitesimal hole is introduced in the body (Soko"owski and Zochowski, 1999). Originally it found applications in the structural optimization community in the so-called bubble method (Eschenauer et al., 1994). In this method, holes are systematically nucleated in strategic locations to both lighten the structure and maintain load integrity. Once the holes are nucleated, traditional Contents lists available at ScienceDirect